Gyrofluid analysis of electron $\beta_e$ effects on collisionless reconnection

The linear and nonlinear evolutions of the tearing instability in a collisionless plasma with a strong guide field are analyzed on the basis of a two-field Hamiltonian gyrofluid model. The model is valid for a low ion temperature and a finite $\beta_e$. The finite $\beta_e$ effect implies a magnetic perturbation along the guide field direction and electron finite Larmor radius effects. A Hamiltonian derivation of the model is presented. A new dispersion relation of the tearing instability is derived for the case $\beta_e=0$ and tested against numerical simulations. For $\beta_e \ll 1$ the equilibrium electron temperature is seen to enhance the linear growth rate, whereas we observe a stabilizing role when electron finite Larmor radius effects become more relevant. In the nonlinear phase, a double "faster-than-exponential" growth is observed, similarly to what occurs in the presence of ion finite Larmor radius effects. Energy transfers are analyzed and the conservation laws associated with the Casimir invariants of the model are also discussed. Numerical simulations seem to indicate that finite $\beta_e$ effects do not produce qualtitative modifications in the structures of the Lagrangian invariants associated with Casimirs of the model.